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Minimal systems of binomial generators and the indispensable complex of a toric ideal

Hara Charalambous·Anargyros Katsabekis·Apostolos Thoma
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Abstract

Let $A=\{{\bf a}_1,...,{\bf a}_m\} \subset \mathbb{Z}^n$ be a vector configuration and $I_A \subset K[x_1,...,x_m]$ its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of different minimal systems of binomial generators of $I_A$. We also prove that generic toric ideals are generated by indispensable binomials. In the second part we associate to $A$ a simplicial complex $Δ_{\ind(A)}$. We show that the vertices of $Δ_{\ind(A)}$ correspond to the indispensable monomials of the toric ideal $I_A$, while one dimensional facets of $Δ_{\ind(A)}$ with minimal binomial $A$-degree correspond to the indispensable binomials of $I_{A}$.

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Hara Charalambous, Anargyros Katsabekis, Apostolos Thoma. 2006-07-11. Minimal systems of binomial generators and the indispensable complex of a toric ideal. https://arxiv.org/abs/math/0607249

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